The fixture saw Mauricio Pochettino come up against the team he has been heavily tipped to take over at the end of the season, but it was his United counterpart - also in the running for the job - who was left celebrating at the final whistle as he passed what was billed as his first major test as interim boss.
The result sees Solskjaer make a slice of history as the first United manager to win his opening six matches in charge, and despite Marcus Rashford scoring the only goal of the game it was David de Gea who proved to be the hero for the visitors.
De Gea made 11 saves in the second half alone to keep Spurs at bay as the hosts were made to rue a plethora of missed chances, squandering the opportunity to move above Manchester City into second place in the process.
The match began at a quick tempo with end-to-end, free-flowing football, and Spurs were the first to come close when Harry Winks latched on to Son Heung-min's through-ball and played a low pass into the box which was too far in front of Harry Kane and Moussa Sissoko, either of whom would have been left with a simple finish.
Winks then fired over from the resulting corner before United hit back with chances of their own as Jesse Lingard sliced a half-volley off target and Rashford then drew a routine stop from Hugo Lloris.
Spurs were struggling to cope with United's counter-attacks, and the hosts almost shot themselves in the foot when Kieran Trippier gave the ball away and allowed Paul Pogba to find Anthony Martial, whose direct running took him into the box before forcing a smart low save from Lloris at the front post.
Kane saw a chance quickly closed down when he was found unmarked inside the area from a corner, but he did have the ball in the back of the net shortly after the half-hour mark only for his close-range finish from Dele Alli's bundled pass across the six-yard box to be correctly ruled out for offside.
It looked as though the two sides would go into the break all square until United sprung into action with just a minute remaining of the first half, capitalising on another misplaced pass from Trippier.
Pogba released Rashford with an exquisite long pass forward, and the United man had the finish to match with a pinpoint low strike across goal and into the bottom far corner.
Spurs ended with the first half without registering a single shot on target, but it was an entirely different story in the second half and Kane began the siege of United's goal by putting his finish too close to De Gea when looking for the same corner that Rashford had found just before half time.
It was the first of a string of saves the Spaniard would need to make in the second half, and another arrived just one minute later when he got down smartly to turn Alli's well-directed header away from the bottom corner.
Such a quick start set the tone for an entertaining second half, and United played their part in that with Pogba forcing two acrobatic saves from compatriot Lloris in quick succession - first from a header and then from a rampaging run into the box which almost ended with him lifting his finish over the keeper and in.
De Gea was by far the busier of the two keepers after the interval, though, and he was called into action again just before the hour mark to make a relatively routine stop from Kane's close-range diving header.
Another Spurs opportunity went begging when Erik Lamela - on as a first-half sub for the injured Sissoko - was unsighted when going for a header inside the area, and United almost doubled their advantage moments later as Martial teed up Pogba, whose deflected drive from the edge of the area was well saved by Lloris.
In a role reversal of the first half it was Spurs who looked most dangerous on the counter for much of the second, and Alli should have put the finishing touch to a particularly devastating break when he put his finish too close to De Gea with only the keeper to beat.
Eriksen then flashed a free kick narrowly wide before De Gea pulled off arguably his best stop of the lot, reacting smartly to keep out Toby Alderweireld's effort on the line with his feet.
Moments later the Spaniard went flying across his goal to pluck a Kane free kick out of the air, and yet another huge Spurs chance went begging just three minutes after that when Alli took Eriksen's pass down inside the area but was then denied by the in-form United shot-stopper.
As impressive as De Gea's performance was, Tottenham's finishing could have been far more clinical and Kane was among the worst culprits. Perhaps the home side's best chance arrived with just four minutes remaining when Fernando Llorente teed up the England captain, only for him to once again show the keeper enough for him to make the save.
Llorente then had a golden chance of his own from inside the area in the 89th minute, but he scuffed his finish straight at De Gea and that proved to be the last of the onslaught as United somehow held on.
The result sees United move level on points with Arsenal in fifth place as Solskjaer surpasses Sir Matt Busby's record for the best start as Red Devils manager, while also seeing his side keep back-to-back clean sheets in the Premier League for the first time this season.
Spurs, meanwhile, miss the chance to close the gap on leaders Liverpool back to six points courtesy of their first defeat of 2019, condemning them to successive home league losses for the first time since May 2015.
TOTTENHAM (4-4-2): Lloris; Trippier, Alderweireld, Vertonghen, Davies; Winks (Llorente 81'), Sissoko (Lamela 43'), Eriksen, Dele; Son, Kane
MAN UTD (4-4-2): De Gea; Young, Lindelof, Jones, Shaw; Matic, Herrera, Lingard (Dalot 83'), Pogba (McTominay 92'); Martial (Lukaku 73'), Rashford